#6438902

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2424
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,15)(4,19)(5,11)(6,12)(7,13)(8,14)(9,18)(10,24)(16,27)(17,26)(20,32)(21,30)(22,29)(23,28)(25,31);(1,3,7,18,8,5,6,17)(2,9,12,15,14,26,13,11)(4,20,23,21,24,31,22,16)(10,27,28,25,19,30,29,32);(1,4,7,23,8,24,6,22)(2,10,12,28,14,19,13,29)(3,16,18,20,5,21,17,31)(9,25,15,30,26,32,11,27);(1,5)(2,11)(3,8)(4,21)(6,18)(7,17)(9,12)(10,27)(13,26)(14,15)(16,24)(19,30)(20,22)(23,31)(25,28)(29,32);(1,6,8,7)(2,12,14,13)(3,17,5,18)(4,22,24,23)(9,15,26,11)(10,28,19,29)(16,31,21,20)(25,30,32,27)

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346098);